Find the equation of the line. It should be of the form $ax + by + c = 0$. Given two points $(x_1, y_2)$ and $(x_2, y_2)$, plug these into that equation. They are on opposite side of the line if $ax_1 + by_1 + c < 0$ and $ax_2 + by_2 + c > 0$, or visa-versa.

$CD$ are on the opposite sides of $AB$ if and only if $\left(\overrightarrow{AB}\times\overrightarrow{AC}\right)\cdot\left(\overrightarrow{AB}\times\overrightarrow{AD}\right)<0$, where $\times$ is cross product and $\cdot$ is dot product.

Writing $A$ and $B$ for the points in question, and $P_1$ and $P_2$ for the points determining the line ...

Compute the "signed" areas of the $\triangle P_1 P_2 A$ and $\triangle P_1 P_2 B$ via the formula (equation 16 here)
$$\frac{1}{2}\left|\begin{array}{ccc}
x_1 & y_1 & 1 \\
x_2 & y_2 & 1 \\
x_3 & y_3 & 1
\end{array}\right|$$
with $(x_3,y_3)$ being $A$ or $B$. The points $A$ and $B$ will be on opposite sides of the line if the areas differ in sign, which indicates that the triangles are being traced-out in different directions (clockwise vs counterclockwise).

You can, of course, ignore the "$1/2$", as it has not affect the sign of the values. Be sure to keep the row-order consistent in the two computations, though.